Finite Group Databasebeta

← #37510168

Subgroups — #37510168

Order 22912.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26039065C211yesmaximal, characteristic
2#26038300C212yesmaximal
3#26038300C211yesmaximal, characteristic
4#26038205C212yesmaximal
5#342×C179C211yesmaximal, characteristic
6C23×Dic179C2211yescharacteristic
7#19153092—22no—
8#19153090—24no—
9C2×C22⋊Dic179C2211yescharacteristic
10C2×C22⋊Dic179C2212yes—
11C179⋊#95—24no—
12C179⋊#95—22no—
13C2×C22⋊C4×C179C2212yes—
14C2×C22⋊C4×C179C2211yescharacteristic
15C2×Dic358—22no—
16C22×Dic179—22no—
17C22×Dic179—21no—
18C22×Dic179—21no—
19C22×Dic179—21no—
20C22×Dic179—22no—
21C22×Dic179D414yes—
22C22×Dic179Q812yescomplemented
23C4⋊Dic179—22no—
24Dic179⋊C4—24no—
25C22⋊Dic179—22no—
26C22⋊Dic179—22no—
27C22⋊Dic179—22no—
28C24×C179C2311yescharacteristic
29C22×C4×C179—22no—
30C22×C4×C179D412yes—
31C22⋊C4×C179—22no—
32C22⋊C4×C179—22no—
33C22⋊C4×C179—22no—
34Dic358—82no—
35C2×Dic179—24no—
36C2×Dic179—24no—
37C2×Dic179—22no—
38C2×Dic179—41no—
39C2×Dic179—42no—
40C2×Dic179—41no—
41C2×Dic179—44no—
42C2×Dic179—42no—
43C2×Dic179—41no—
44C23×C179—42no—
45C23×C179C2×D412yes—
46C23×C179C2×D411yescharacteristic
47C23×C179C2×Q811yescharacteristic
48C23×C179#4111yescharacteristic
49C23×C179#4112yes—
50C2×C4×C179—22no—
51C2×C4×C179—42no—
52C2×C4×C179—44no—
53C2×C4×C179—42no—
54C22×C179—24no—
55C22×C179—22no—
56C22×C179—41no—
57C22×C179—41no—
58C22×C179—42no—
59C22×C179#12011yescharacteristic
60C22×C179C22⋊Q812yes—
61C22×C179C22⋊Q811yescharacteristic
62C22×C179#12412yes—
63C22×C179#12411yescharacteristic
64C4×C179—42no—
65C4×C179—82no—
66Dic179—42no—
67Dic179—44no—
68Dic179—81no—
69C2×C179—21no—
70C2×C179—22no—
71C2×C179—22no—
72C2×C179—21no—
73C2×C179#39311yescharacteristic
74C2×C179#45812yes—
75C179#203011yescharacteristic, complemented, Sylow
76#2030—1791nomaximal, Sylow
77#543—1791no—
78#381—1792no—
79#381—1791no—
80#342—1792no—
81#342D17911yescharacteristic
82C23×C4—1791no—
83C22⋊Q8—3584no—
84C22⋊Q8—3582no—
85C2×C22⋊C4—1791no—
86C2×C22⋊C4—1792no—
87C2×C22⋊C4C2×D17912yes—
88C2×C22⋊C4C2×D17911yescharacteristic
89#95—3584no—
90#95—3582no—
91C24C22×D17911yescharacteristic
92C2×Q8—3582no—
93C22×C4—22no—
94C22×C4—3582no—
95C22×C4—3581no—
96C22×C4—3581no—
97C22×C4—3582no—
98C22×C4—3581no—
99C22×C4—1792no—
100C22×C4—1794no—
101C22×C4D71612yes—
102C4⋊C4—3584no—
103C4⋊C4—3582no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—3582no—
108C22⋊C4—3582no—
109C22⋊C4—3582no—
110C23—42no—
111C23D4×D17912yes—
112C23C2×D71611yescharacteristic
113C23Q8×D17911yescharacteristic
114C23#1566825811yescharacteristic
115C23#1566825912yes—
116Q8—14322no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—3584no—
122C2×C4—3584no—
123C2×C4—3582no—
124C2×C4—7162no—
125C2×C4—7161no—
126C2×C4—7161no—
127C2×C4—7162no—
128C2×C4—7161no—
129C2×C4—7164no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#1915308311yescharacteristic
136C22#1915309112yes—
137C22#1915309311yescharacteristic
138C22#1915309912yes—
139C22#1915310111yescharacteristic
140C4—42no—
141C4—82no—
142C4—14321no—
143C4—7162no—
144C4—7164no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#2603835211yescharacteristic
150C2#2603863012yes—
subgroupG/H
centerC22#19153083
commutator, prime radical, cosocleC24×C179C23
FrattiniC24C22×D179
Fitting#342×C179C2
socleC22×C179#120
hypercenter#342D179
hypocenterC179#2030
LCSG ▷ C24×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
22. C22×Dic179Q8semidirect
75. C179#2030semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260390652
#260383002
#260382052
#342×C1792
#2030179